Page "3-sphere" Paragraph 16

Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere ( unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point ).

As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere that reaches its maximal size when the hyperplane cuts right through the " equator " of the 3-sphere.

Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.